IODS course project
Yu Ren
Introduction to Open Data Science - Course Project
About the project
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you should use R Markdown syntax.
The text continues here.
I just solved the problem of uploading Rstudio to Github before writing chapter 1, thanks to the help of my classmate, I’m very happy to have solved this trouble!
I’m really looking forward to this course, after having studied QSR I’m really looking forward to continuing my in-depth study of R and analytics, and I also really like Kimmo’s course, so I’ve chosen this course.
The R for Health Data Science book was something I had already come across in the QSR and the second time through I was able to get to grips with some of the necessary commands in R. The parts that were difficult to understand before were now understandable and this was a very necessary process. The link: https://github.com/YuRenn/IODS-project
Describe the work you have done this week and summarize your learning. Data wrangling and analysis
- Describe your work and results clearly.
- Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods.
- Assume the reader has no previous knowledge of your data or the more advanced methods you are using.
1.read date set
learning2014 <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/learning2014.txt",
sep = ",", header = T)- Show a graphical overview of the data and show summaries of the variables in the data.
View(learning2014)
dim(learning2014)## [1] 166 7str(learning2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : chr "F" "M" "F" "M" ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...This data set is from international survey of Approaches to Learning, made possible by Teachers’ Academy funding for KV in 2013-2015. Before data wrangling, the data set has 183 observations and 60 variables. Then I did a data wrangling, and choose the variable of attitude, which is a sum of 10 questions related to students attitude towards statistics. In the dataset of learning2014, it has 166 observations and 7 variables, they are gender, age,attitude(Global attitude toward statistics), deep(deep learning), stra(strategical learning), surf(surface learning), and points(exam points).
3.Choose three variables as explanatory variables and fit a regression model where exam points is the target (dependent, outcome) variable. 3.1First, visualize the data set and create a plot matrix with ggpairs()
library(GGally)## Loading required package: ggplot2## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)
ggpairs(learning2014, lower = list(combo = wrap("facethist", bins = 20)))
3.2Choose attitude, points and surf as variables and fit a regression model
fit <- learning2014 %>%
lm(points ~ attitude + surf, data = .)
summary(fit)##
## Call:
## lm(formula = points ~ attitude + surf, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.277 -3.236 0.386 3.977 10.642
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.1196 3.1271 4.515 1.21e-05 ***
## attitude 3.4264 0.5764 5.944 1.63e-08 ***
## surf -0.7790 0.7956 -0.979 0.329
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 163 degrees of freedom
## Multiple R-squared: 0.1953, Adjusted R-squared: 0.1854
## F-statistic: 19.78 on 2 and 163 DF, p-value: 2.041e-08The result shows that expect surf, the p-values for the other variables are both significant and predictive. adjust R-squared shows that the model explains 18.54%.
3.3 remove the variable of surf, and fit the model again.
fit2 <- learning2014 %>%
lm(points ~ attitude, data = .)
summary(fit2)##
## Call:
## lm(formula = points ~ attitude, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.6372 1.8303 6.358 1.95e-09 ***
## attitude 3.5255 0.5674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09There is a strong correlation between the two variables and the model is a valid model. Adjusted R-squared indicates that the model can explain 18.54 of the variance, which is only 0.01% higher than fit1. A coefficient of 3.53 for attitude means that for every 1 increase in attitude, the test points will increase by 3.53.
4.Model diagnostic
library(ggfortify)
autoplot(fit2)
The Residuals vs fitted plot shows the data distributed up and down
along the blue line with no clear trend, indicating a linear
relationship.
prepare some packages
library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ tibble 3.1.8 ✔ dplyr 1.0.10
## ✔ tidyr 1.2.1 ✔ stringr 1.4.1
## ✔ readr 2.1.3 ✔ forcats 0.5.2
## ✔ purrr 0.3.4
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library(dplyr)- read the joined student alcohol consumption data
alc <- read_csv("alc.csv")## Rows: 370 Columns: 35
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (17): school, sex, address, famsize, Pstatus, Mjob, Fjob, reason, guardi...
## dbl (17): age, Medu, Fedu, traveltime, studytime, famrel, freetime, goout, D...
## lgl (1): high_use
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.1.2 Print out the names of the variables in the data and describe the data set briefly
glimpse(alc)## Rows: 370
## Columns: 35
## $ school <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP",…
## $ sex <chr> "F", "F", "F", "F", "F", "M", "M", "F", "M", "M", "F", "F",…
## $ age <dbl> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 15, 15,…
## $ address <chr> "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U",…
## $ famsize <chr> "GT3", "GT3", "LE3", "GT3", "GT3", "LE3", "LE3", "GT3", "LE…
## $ Pstatus <chr> "A", "T", "T", "T", "T", "T", "T", "A", "A", "T", "T", "T",…
## $ Medu <dbl> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3, 3, 4,…
## $ Fedu <dbl> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3, 2, 3,…
## $ Mjob <chr> "at_home", "at_home", "at_home", "health", "other", "servic…
## $ Fjob <chr> "teacher", "other", "other", "services", "other", "other", …
## $ reason <chr> "course", "course", "other", "home", "home", "reputation", …
## $ guardian <chr> "mother", "father", "mother", "mother", "father", "mother",…
## $ traveltime <dbl> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1,…
## $ studytime <dbl> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 1, 1,…
## $ schoolsup <chr> "yes", "no", "yes", "no", "no", "no", "no", "yes", "no", "n…
## $ famsup <chr> "no", "yes", "no", "yes", "yes", "yes", "no", "yes", "yes",…
## $ activities <chr> "no", "no", "no", "yes", "no", "yes", "no", "no", "no", "ye…
## $ nursery <chr> "yes", "no", "yes", "yes", "yes", "yes", "yes", "yes", "yes…
## $ higher <chr> "yes", "yes", "yes", "yes", "yes", "yes", "yes", "yes", "ye…
## $ internet <chr> "no", "yes", "yes", "yes", "no", "yes", "yes", "no", "yes",…
## $ romantic <chr> "no", "no", "no", "yes", "no", "no", "no", "no", "no", "no"…
## $ famrel <dbl> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5, 5, 3,…
## $ freetime <dbl> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3, 5, 1,…
## $ goout <dbl> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2, 5, 3,…
## $ Dalc <dbl> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1,…
## $ Walc <dbl> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 3,…
## $ health <dbl> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4, 5, 5,…
## $ failures <dbl> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ paid <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes", …
## $ absences <dbl> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3, 9, 5,…
## $ G1 <dbl> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 14, 16,…
## $ G2 <dbl> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, 15, 16…
## $ G3 <dbl> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12, 16, 1…
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5, 1.0,…
## $ high_use <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FALS…The dataset has 370 observations and 35 variables. I will explain some meanings of variables: famsize-family size; Pstatus - parent’s cohabitation status; Medu - mother’s education; Fedu - father’s education; Mjob - mother’s job; Fjob - father’s job; reason - reason to choose this school; guardian - student’s guardian; schoolsup - extra educational support; famsup - family educational support; activities - extra-curricular activities; nursery - attended nursery school; higher - wants to take higher education; internet - Internet access at home; romantic - with a romantic relationship; famrel - quality of family relationships; freetime - free time after school; goout - going out with friends; Dalc - workday alcohol consumption; Walc - weekend alcohol consumption; health - current health status; absences - number of school absences; G1 - first period grade; G2 - second period grade; G3 - final grade.
- Choose 4 variables and present personal hypothesis The four variables I have chosen are goout - going out with friends, freetime - free time after school，famrel - quality of family relationships and absences - number of school absences. I suspect that these four variables would be related to student drinking; having free time after school and going out with friends multiple times may increase the frequency of drinking, the quality of family relationships may also influence drinking status, and there may be a relationship between absence from class and drinking.
3.Explore the distributions of variables and their relationships with alcohol consumption
alc %>% group_by(freetime) %>% summarise(count = n())## # A tibble: 5 × 2
## freetime count
## <dbl> <int>
## 1 1 17
## 2 2 60
## 3 3 152
## 4 4 105
## 5 5 36alc %>% group_by(goout) %>% summarise(count = n())## # A tibble: 5 × 2
## goout count
## <dbl> <int>
## 1 1 22
## 2 2 97
## 3 3 120
## 4 4 78
## 5 5 53alc %>% group_by(famrel) %>% summarise(count = n())## # A tibble: 5 × 2
## famrel count
## <dbl> <int>
## 1 1 8
## 2 2 18
## 3 3 64
## 4 4 180
## 5 5 100alc %>% group_by(absences) %>% summarise(count = n())## # A tibble: 26 × 2
## absences count
## <dbl> <int>
## 1 0 63
## 2 1 50
## 3 2 56
## 4 3 38
## 5 4 35
## 6 5 22
## 7 6 21
## 8 7 12
## 9 8 20
## 10 9 11
## # … with 16 more rowsThe distribution of the freetime and goout variables is more in the middle, with the family relationship variables mostly distributed between 3 and 5, indicating that the majority of students have good family relationships; classroom absences, the majority of students are absent 5 times or less.
3.1 explore alcohol high use and freetime
p1 <- alc %>%
ggplot(aes(x = freetime, fill = high_use)) +
geom_bar(position = "fill", color ="white") +
labs(x = "free time after school", y = "alcohol high-user",
title = "Proportion of alcohol high-use by freetime")+
theme(legend.position = "bottom")+
scale_fill_discrete(labels = c("FALSE" = "Non-high-user",
"TRUE" = "high-user"))+
scale_fill_brewer(palette = "")## Warning in pal_name(palette, type): Unknown palette## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.p1
According to the proportions shown in the bar chart, the proportion of
alcohol use increases gradually as leisure time after school
increases.
3.2 explore alcohol high use and goout
p2 <- alc %>%
ggplot(aes(x = goout, fill = high_use)) +
geom_bar(position = "fill", color ="white") +
labs(x = "going out with friends", y = "alcohol high-user",
title = "Proportion of alcohol high-use by goout")+
theme(legend.position = "bottom")+
scale_fill_discrete(labels = c("FALSE" = "Non-high-user",
"TRUE" = "high-user"))+
scale_fill_brewer(palette = "")## Warning in pal_name(palette, type): Unknown palette## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.p2
According to the bar scale, the proportion of alcohol consumption
increases as the number of outings with friends increases, particularly
at level 4 of the outing degree, where the proportion increases
substantially.
3.3explore alcohol high use and famrel
p3 <- alc %>%
ggplot(aes(x = famrel, fill = high_use)) +
geom_bar(position = "fill", color ="white") +
labs(x = "quality of family relationships", y = "alcohol high-user",
title = "Proportion of alcohol high-use by family relationship")+
theme(legend.position = "bottom")+
scale_fill_discrete(labels = c("FALSE" = "Non-high-user",
"TRUE" = "high-user"))+
scale_fill_brewer(palette = "")## Warning in pal_name(palette, type): Unknown palette## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.p3
“famrel - quality of family relationships (numeric: from 1 - very bad to
5 - excellent)” Family relationships start at level 3, with better
family relationships being accompanied by a lower proportion of alcohol
consumption. The highest proportion of alcohol consumption is found when
family relationships are poor.
3.4 explore alcohol high use and absences
p4 <- alc %>%
ggplot(aes(x = absences, fill = high_use)) +
geom_bar(position = "fill", color ="white") +
labs(x = " number of school absences (numeric: from 0 to 93)", y = "alcohol high-user",
title = "Proportion of alcohol high-use by number of school absences")+
theme(legend.position = "bottom")+
scale_fill_discrete(labels = c("FALSE" = "Non-high-user",
"TRUE" = "high-user"))+
scale_fill_brewer(palette = "")## Warning in pal_name(palette, type): Unknown palette## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.p4
As the number of absences from school increases, the proportion of
alcohol consumption increases with it. The bars show much the same
results as I expected, the only differences being family relationships
and alcohol consumption rates, with the worst families not having the
highest rates of alcohol consumption and the second worst families
having the highest rates of alcohol consumption.
4.logistic regression model 4.1
m1 <- glm(high_use ~ freetime + goout + famrel + absences, data = alc, family = "binomial")
summary(m1)##
## Call:
## glm(formula = high_use ~ freetime + goout + famrel + absences,
## family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.8393 -0.7676 -0.5200 0.9080 2.3909
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.7391 0.6975 -3.927 8.6e-05 ***
## freetime 0.2283 0.1393 1.639 0.101292
## goout 0.7103 0.1252 5.673 1.4e-08 ***
## famrel -0.4019 0.1387 -2.898 0.003754 **
## absences 0.0761 0.0220 3.459 0.000543 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 380.08 on 365 degrees of freedom
## AIC: 390.08
##
## Number of Fisher Scoring iterations: 4There is a highly significant relationship between the proportion of increased drinking and family relationships, the frequency of going out with friends and absence from class, but a non-significant p-value between this and free time.
4.2 From coefficients to odds ratios
m1 <- glm(high_use ~ freetime + goout + famrel + absences, data = alc, family = "binomial")
# compute odds ratios (OR)
OR <- coef(m1) %>% exp
OR## (Intercept) freetime goout famrel absences
## 0.06462506 1.25649062 2.03450512 0.66907685 1.07907060# compute confidence intervals (CI)
confint(m1)## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) -4.14545349 -1.4024140
## freetime -0.04305995 0.5046433
## goout 0.47111595 0.9630772
## famrel -0.67730666 -0.1316918
## absences 0.03387575 0.1213892CI <- exp(confint(m1))## Waiting for profiling to be done...CI## 2.5 % 97.5 %
## (Intercept) 0.01583625 0.2460024
## freetime 0.95785396 1.6563946
## goout 1.60178070 2.6197457
## famrel 0.50798332 0.8766111
## absences 1.03445607 1.1290642# print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.06462506 0.01583625 0.2460024
## freetime 1.25649062 0.95785396 1.6563946
## goout 2.03450512 1.60178070 2.6197457
## famrel 0.66907685 0.50798332 0.8766111
## absences 1.07907060 1.03445607 1.1290642According to the model, the probability of drinking increases 1.26 times for every doubling of free time, 2.03 times for more frequent outings with friends, 0.67 times for every unit of improvement in family relations, and 1.08 times for every increase in absenteeism from school.
- Explore the predictive power of the model 5.1 Binary predictions（1）
prob <- predict(m1, type = "response")
library(dplyr)
alc <- mutate(alc, probability = prob)
alc <- mutate(alc, prediction = probability > 0.5)
table(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 239 20
## TRUE 67 44Of the 370 participants, there are 259 non-alcoholics, of which the model correctly predicted 239 (92.28%); of the 111 alcoholics, the model successfully predicted 44 (39.64%).
5.2 Binary predictions (2)
library(dplyr); library(ggplot2)
# initialize a plot of 'high_use' versus 'probabilities' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use))
# define the geom as points and draw the plot
g+aes(color = prediction, shape = prediction) + geom_point()
# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 239 20
## TRUE 67 446.Bonus
# define a loss function (mean prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}training.error.full <- loss_func(alc$high_use, alc$probability)
training.error.full## [1] 0.2351351Prediction error rate of 23.51%.
- Load the Boston data from the MASS package
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(corrplot)## corrplot 0.92 loadedlibrary(tidyverse)
data("Boston")1,1 explore the structure and the dimensions of the data
#explore the structure
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...#explore the dimensions#
dim(Boston)## [1] 506 14This dataset has 506 observations and 14 variables. And the variable crim = per capita crime rate by town; zn = proportion of residential land zoned for lots over 25,000 sq.ft; indus = proportion of non-retail business acres per town; chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise); nox = nitrogen oxides concentration (parts per 10 million); rm = average number of rooms per dwelling; dis = weighted mean of distances to five Boston employment centres; rad = index of accessibility to radial highways; tax = full-value property-tax rate per $10,000; ptratio = pupil-teacher ratio by town; black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town; lstat = lower status of the population (percent); medv = median value of owner-occupied homes in $1000s.
2.Show a graphical overview of the data and show summaries of the variables in the data.
# plot matrix of the variables
pairs(Boston)
# calculate the correlation matrix and round it
cor_matrix <- cor(Boston)
# print the correlation matrix
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax ptratio
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431 0.2899456
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018 0.3832476
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320 0.1889327
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783 -0.3555015
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559 0.2615150
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819 0.4647412
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000 0.4608530
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304 1.0000000
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341 0.3740443
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593 -0.5077867
## black lstat medv
## crim -0.38506394 0.4556215 -0.3883046
## zn 0.17552032 -0.4129946 0.3604453
## indus -0.35697654 0.6037997 -0.4837252
## chas 0.04878848 -0.0539293 0.1752602
## nox -0.38005064 0.5908789 -0.4273208
## rm 0.12806864 -0.6138083 0.6953599
## age -0.27353398 0.6023385 -0.3769546
## dis 0.29151167 -0.4969958 0.2499287
## rad -0.44441282 0.4886763 -0.3816262
## tax -0.44180801 0.5439934 -0.4685359
## ptratio -0.17738330 0.3740443 -0.5077867
## black 1.00000000 -0.3660869 0.3334608
## lstat -0.36608690 1.0000000 -0.7376627
## medv 0.33346082 -0.7376627 1.0000000# visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method="circle")
From the correlation matrix, it is possible to look for relationships
between different variables, using the variable dis as an example, the
weighted average of distances to the five employment centres in Boston
and the variable indus (proportion of non-retail commercial acres in
each town) show a very strong negative correlation; with nox (nitrogen
oxide concentration) also a strong negative correlation; and with age
again a strong negative correlation. There is a relatively strong
positive correlation with zn (the proportion of residential land over
25,000 square feet).
#summaries of the variables in the data
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00- Standardize the dataset and print out summaries of the scaled data.
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
boston_scaled## crim zn indus chas nox rm
## 1 -0.419366929 0.28454827 -1.28663623 -0.2723291 -0.14407485 0.413262920
## 2 -0.416926670 -0.48724019 -0.59279438 -0.2723291 -0.73953036 0.194082387
## 3 -0.416928995 -0.48724019 -0.59279438 -0.2723291 -0.73953036 1.281445551
## 4 -0.416338404 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.015297761
## 5 -0.412074053 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.227362043
## 6 -0.416631374 -0.48724019 -1.30558569 -0.2723291 -0.83445805 0.206891639
## 7 -0.409837246 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8 -0.403296561 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9 -0.395543302 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10 -0.400333140 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11 -0.393956378 0.04872402 -0.47618230 -0.2723291 -0.26489191 0.131459378
## 12 -0.406444832 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13 -0.409198989 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14 -0.346886928 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15 -0.345933611 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16 -0.347162460 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17 -0.297573695 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18 -0.328932014 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19 -0.326780075 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20 -0.335721492 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21 -0.274570851 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22 -0.321045059 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23 -0.276816959 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24 -0.305188606 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25 -0.332877817 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26 -0.322382028 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27 -0.341986646 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28 -0.308985598 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29 -0.330235268 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.299402903
## 30 -0.303558666 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.554164692
## 31 -0.288635766 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32 -0.262604396 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33 -0.258736486 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34 -0.286204807 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35 -0.232598159 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36 -0.412641393 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37 -0.408773484 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38 -0.410784750 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39 -0.399750686 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40 -0.416889467 2.72854505 -1.19334657 -0.2723291 -1.09335175 0.441727925
## 41 -0.416196569 2.72854505 -1.19334657 -0.2723291 -1.09335175 1.052302267
## 42 -0.405285738 -0.48724019 -0.61611679 -0.2723291 -0.92075595 0.690796712
## 43 -0.403651148 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44 -0.401574777 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45 -0.405837965 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46 -0.400172703 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47 -0.398203290 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48 -0.393447167 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49 -0.390587216 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50 -0.394551620 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51 -0.409786092 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52 -0.415059564 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53 -0.413870242 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
## 54 -0.414310861 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55 -0.418520570 2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56 -0.418577536 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 57 -0.417712575 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 58 -0.418436864 3.80047346 -1.43094368 -0.2723291 -1.24005818 0.756266222
## 59 -0.402145605 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60 -0.408094536 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61 -0.402742009 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62 -0.400138988 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63 -0.407281891 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.243896145
## 64 -0.405395021 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.679410711
## 65 -0.417833484 0.26310970 -1.42219777 -0.2723291 -1.19604625 1.166162284
## 66 -0.415934988 2.94293073 -1.13212523 -0.2723291 -1.35224545 0.007636609
## 67 -0.415010735 2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68 -0.413371495 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69 -0.404344047 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70 -0.405202032 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71 -0.409840734 -0.48724019 -0.04763292 -0.2723291 -1.22279860 0.188389387
## 72 -0.401644532 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73 -0.409447781 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74 -0.397385995 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75 -0.410921935 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76 -0.409043203 -0.48724019 0.24681257 -0.2723291 -1.01568364 0.001943608
## 77 -0.408297988 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78 -0.409979081 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79 -0.413537744 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80 -0.410351107 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81 -0.415319982 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.629596953
## 82 -0.414914241 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.475885930
## 83 -0.415847794 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.024715612
## 84 -0.415973353 0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85 -0.414220179 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 86 -0.413434274 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.491541682
## 87 -0.414070206 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88 -0.411788058 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89 -0.413521468 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 90 -0.413937672 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.130581029
## 91 -0.414656148 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.188389387
## 92 -0.415530409 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.171310384
## 93 -0.415215350 0.71331963 0.56895343 -0.2723291 -0.78267931 0.223970642
## 94 -0.416759258 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95 -0.415109555 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96 -0.405913532 -0.48724019 -1.20209248 -0.2723291 -0.94664532 0.484425431
## 97 -0.406727340 -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98 -0.406054205 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.539598741
## 99 -0.410583624 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.185209437
## 100 -0.412126370 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 101 -0.402818740 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.629596953
## 102 -0.406811046 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.706452465
## 103 -0.393506459 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.171310384
## 104 -0.395500287 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.403872039 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.404683521 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.400198280 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.404852095 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.405218308 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.269514649
## 110 -0.389452536 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.407553935 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.408378206 -0.48724019 -0.16424500 -0.2723291 -0.06640675 0.612517950
## 113 -0.405768210 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.394278413 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.403556979 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.400182004 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.404804429 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.402549021 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.404920687 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.403272146 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.412081029 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 122 -0.411771782 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.399412952
## 123 -0.409290833 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.460612711
## 124 -0.402618775 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.610053984
## 125 -0.408651413 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.577319229
## 126 -0.400451723 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.425031456
## 127 -0.375069074 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.955903786
## 128 -0.389973373 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.842043769
## 129 -0.382267781 -0.48724019 1.56744433 -0.2723291 0.59808708 0.208314890
## 130 -0.317649158 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.921745781
## 131 -0.380566923 -0.48724019 1.56744433 -0.2723291 0.59808708 0.246742645
## 132 -0.281412645 -0.48724019 1.56744433 -0.2723291 0.59808708 0.058873617
## 133 -0.351503540 -0.48724019 1.56744433 -0.2723291 0.59808708 0.124343127
## 134 -0.381757407 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.658444491
## 135 -0.306613931 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.750955755
## 136 -0.355255192 -0.48724019 1.56744433 -0.2723291 0.59808708 0.071682869
## 137 -0.382592141 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.487654465
## 138 -0.379140436 -0.48724019 1.56744433 -0.2723291 0.59808708 0.241049645
## 139 -0.391060387 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.608630733
## 140 -0.356796775 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.190195170
## 141 -0.386282176 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.157460416
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## 338 -0.416566270 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.554547225
## 339 -0.416258185 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.321134190
## 340 -0.413710969 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.426454706
## 341 -0.412950640 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.450649960
## 342 -0.418589162 1.01345959 -1.40179066 -0.2723291 -0.97253469 1.361147563
## 343 -0.417197552 -0.48724019 -1.34785757 -0.2723291 -0.31667065 0.363449163
## 344 -0.417145235 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.585476196
## 345 -0.416556969 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.838814735
## 346 -0.416482564 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 347 -0.412937852 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.550277475
## 348 -0.417927653 3.15731641 -1.01842846 -0.2723291 -1.08472196 0.329291158
## 349 -0.418356646 2.94293073 -1.33036576 -0.2723291 -1.03294322 0.498657933
## 350 -0.416731356 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.931325998
## 351 -0.412880885 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.292286652
## 352 -0.410859155 2.08538800 -1.37701059 -0.2723291 -1.24005818 0.418955921
## 353 -0.411679938 2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 354 -0.418114829 3.37170210 -1.32890811 -0.2723291 -1.24868797 0.631020203
## 355 -0.415101417 2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.884741275
## 356 -0.407709721 2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.496193967
## 357 0.624240921 -0.48724019 1.01499462 3.6647712 1.85803641 -0.103376907
## 358 0.027457444 -0.48724019 1.01499462 3.6647712 1.85803641 0.157077882
## 359 0.184646645 -0.48724019 1.01499462 3.6647712 1.85803641 -0.224353176
## 360 0.075310474 -0.48724019 1.01499462 -0.2723291 1.85803641 -0.245701929
## 361 0.107933683 -0.48724019 1.01499462 -0.2723291 1.85803641 0.161347633
## 362 0.025962364 -0.48724019 1.01499462 -0.2723291 1.85803641 -0.047870149
## 363 0.007521491 -0.48724019 1.01499462 -0.2723291 1.85803641 -1.313139590
## 364 0.070785706 -0.48724019 1.01499462 3.6647712 1.85803641 -0.685486245
## 365 -0.016188203 -0.48724019 1.01499462 3.6647712 1.40928733 3.551529643
## 366 0.109555485 -0.48724019 1.01499462 -0.2723291 1.40928733 -3.876413226
## 367 0.009699007 -0.48724019 1.01499462 -0.2723291 1.40928733 -1.881016425
## 368 1.151964713 -0.48724019 1.01499462 -0.2723291 0.65849561 -3.446591661
## 369 0.149356473 -0.48724019 1.01499462 -0.2723291 0.65849561 -1.871053674
## 370 0.239079888 -0.48724019 1.01499462 3.6647712 0.65849561 0.566973944
## 371 0.340082672 -0.48724019 1.01499462 3.6647712 0.65849561 1.040916265
## 372 0.653228737 -0.48724019 1.01499462 -0.2723291 0.65849561 -0.097683907
## 373 0.541033778 -0.48724019 1.01499462 3.6647712 0.97779784 -0.583012230
## 374 0.871305835 -0.48724019 1.01499462 -0.2723291 0.97779784 -1.962141687
## 375 1.730465429 -0.48724019 1.01499462 -0.2723291 0.97779784 -3.055197852
## 376 1.859616644 -0.48724019 1.01499462 -0.2723291 1.00368721 1.463621579
## 377 1.357253412 -0.48724019 1.01499462 -0.2723291 1.00368721 0.518583436
## 378 0.721959412 -0.48724019 1.01499462 -0.2723291 1.00368721 0.724954717
## 379 2.329195069 -0.48724019 1.01499462 -0.2723291 1.00368721 0.135729129
## 380 1.657048387 -0.48724019 1.01499462 -0.2723291 1.00368721 -0.087721155
## 381 9.924109610 -0.48724019 1.01499462 -0.2723291 1.00368721 0.972600255
## 382 1.425427210 -0.48724019 1.01499462 -0.2723291 1.00368721 0.370565414
## 383 0.647964566 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.065494052
## 384 0.509089517 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.088266056
## 385 1.914932287 -0.48724019 1.01499462 -0.2723291 1.25395112 -2.727850303
## 386 1.534407630 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.434115858
## 387 2.415877170 -0.48724019 1.01499462 -0.2723291 1.25395112 -2.323647242
## 388 2.206996093 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.828356167
## 389 1.246308229 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.999146193
## 390 0.527604795 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.273288584
## 391 0.389305224 -0.48724019 1.01499462 -0.2723291 1.25395112 -0.813578764
## 392 0.195258692 -0.48724019 1.01499462 -0.2723291 1.25395112 -0.332520192
## 393 0.925923929 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.777119159
## 394 0.584922404 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.130418661
## 395 1.133084385 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.565933227
## 396 0.593291831 -0.48724019 1.01499462 -0.2723291 1.19354259 0.265244898
## 397 0.262572179 -0.48724019 1.01499462 -0.2723291 1.19354259 0.171310384
## 398 0.471833420 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.765188257
## 399 4.038608880 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.183623820
## 400 0.732778398 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.615746985
## 401 2.491712382 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.423608206
## 402 1.234973056 -0.48724019 1.01499462 -0.2723291 1.19354259 0.083068871
## 403 0.695478123 -0.48724019 1.01499462 -0.2723291 1.19354259 0.169887134
## 404 2.463298882 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.331641842
## 405 4.408007629 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.072610303
## 406 7.476247076 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.856276271
## 407 1.988326078 -0.48724019 1.01499462 -0.2723291 0.90012973 -3.055197852
## 408 0.969311483 -0.48724019 1.01499462 -0.2723291 0.90012973 -0.963020037
## 409 0.440661113 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.950210785
## 410 1.258468834 -0.48724019 1.01499462 -0.2723291 0.36508275 0.807503230
## 411 5.524853484 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.750955755
## 412 1.213407163 -0.48724019 1.01499462 -0.2723291 0.36508275 0.529969438
## 413 1.766830989 -0.48724019 1.01499462 -0.2723291 0.36508275 -2.357805247
## 414 2.911369543 -0.48724019 1.01499462 -0.2723291 0.36508275 -1.607752384
## 415 4.898256758 -0.48724019 1.01499462 -0.2723291 1.19354259 -2.512939520
## 416 1.682381045 -0.48724019 1.01499462 -0.2723291 1.07272553 0.212584640
## 417 0.839462719 -0.48724019 1.01499462 -0.2723291 1.07272553 0.707875715
## 418 2.595705326 -0.48724019 1.01499462 -0.2723291 1.07272553 -1.395688102
## 419 8.128839131 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.466305712
## 420 0.953174847 -0.48724019 1.01499462 -0.2723291 1.40928733 0.767652224
## 421 0.868899291 -0.48724019 1.01499462 -0.2723291 1.40928733 0.179849885
## 422 0.396331868 -0.48724019 1.01499462 -0.2723291 1.40928733 -0.396566452
## 423 0.980600153 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.906090028
## 424 0.399567334 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.258511181
## 425 0.602054210 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.024219796
## 426 1.423787970 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.553123975
## 427 1.003735531 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.637095738
## 428 3.958402360 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.117609410
## 429 0.436385137 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.130418661
## 430 0.665620696 -0.48724019 1.01499462 -0.2723291 1.07272553 0.135729129
## 431 0.567177918 -0.48724019 1.01499462 -0.2723291 0.25289548 0.090185122
## 432 0.749723028 -0.48724019 1.01499462 -0.2723291 0.25289548 0.780461476
## 433 0.329071860 -0.48724019 1.01499462 -0.2723291 0.25289548 0.199775388
## 434 0.228743373 -0.48724019 1.01499462 -0.2723291 1.36613839 0.215431141
## 435 1.197444914 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.109069908
## 436 0.877386138 -0.48724019 1.01499462 -0.2723291 1.59914271 0.490118432
## 437 1.256434316 -0.48724019 1.01499462 -0.2723291 1.59914271 0.251012396
## 438 1.344372005 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.188771920
## 439 1.170089364 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.497617217
## 440 0.671635895 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 441 2.143519126 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.664137492
## 442 0.710413811 -0.48724019 1.01499462 -0.2723291 1.59914271 0.172733634
## 443 0.238660196 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.093414156
## 444 0.738590145 -0.48724019 1.01499462 -0.2723291 1.59914271 0.285170401
## 445 1.068270448 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.612900484
## 446 0.820582390 -0.48724019 1.01499462 -0.2723291 1.59914271 0.248165896
## 447 0.310937908 -0.48724019 1.01499462 -0.2723291 1.59914271 0.080222370
## 448 0.733743341 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.047870149
## 449 0.664481366 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 450 0.454858563 -0.48724019 1.01499462 -0.2723291 1.36613839 0.188389387
## 451 0.360888236 -0.48724019 1.01499462 -0.2723291 1.36613839 0.660908458
## 452 0.212475366 -0.48724019 1.01499462 -0.2723291 1.36613839 0.527122938
## 453 0.171672232 -0.48724019 1.01499462 -0.2723291 1.36613839 0.017599361
## 454 0.538806271 -0.48724019 1.01499462 -0.2723291 1.36613839 1.577481596
## 455 0.685935651 -0.48724019 1.01499462 -0.2723291 1.36613839 0.631020203
## 456 0.132400217 -0.48724019 1.01499462 -0.2723291 1.36613839 0.342100410
## 457 0.122688009 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.439263958
## 458 0.533282845 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.496193967
## 459 0.481158489 -0.48724019 1.01499462 -0.2723291 1.36613839 0.023292362
## 460 0.370589982 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.289822685
## 461 0.139347806 -0.48724019 1.01499462 -0.2723291 1.36613839 0.592592447
## 462 0.009252575 -0.48724019 1.01499462 -0.2723291 1.36613839 0.130036128
## 463 0.353587222 -0.48724019 1.01499462 -0.2723291 1.36613839 0.046064365
## 464 0.256654638 -0.48724019 1.01499462 -0.2723291 1.36613839 0.325021407
## 465 0.491283414 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.107646658
## 466 -0.052307295 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.748109254
## 467 0.018770633 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 468 0.094024554 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 469 1.390700891 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.510426469
## 470 1.099985680 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.813578764
## 471 0.085480740 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.167423167
## 472 0.049396526 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.005213430 -0.48724019 1.01499462 -0.2723291 0.21837632 0.216854391
## 474 0.120137304 -0.48724019 1.01499462 -0.2723291 0.51178918 0.989679257
## 475 0.516449822 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.220628326
## 476 0.323150830 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.174539418
## 477 0.146239592 -0.48724019 1.01499462 -0.2723291 0.51178918 0.283747151
## 478 1.326491497 -0.48724019 1.01499462 -0.2723291 0.51178918 -1.395688102
## 479 0.769568300 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.141804663
## 480 1.246308229 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.079181654
## 481 0.256987136 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 0.243520951 -0.48724019 1.01499462 -0.2723291 -0.19585359 0.662331708
## 483 0.246192564 -0.48724019 1.01499462 -0.2723291 -0.19585359 1.104962525
## 484 -0.092441945 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.143573456 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.588705230
## 486 0.006992516 -0.48724019 1.01499462 -0.2723291 0.24426569 0.038948114
## 487 0.241610829 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.242855428
## 488 0.142084524 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.540314723
## 489 -0.402562972 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.182200570
## 490 -0.398783418 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.239130578
## 491 -0.395982758 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.695993897
## 492 -0.407808541 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 493 -0.407159820 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 494 -0.399952975 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.822118266
## 495 -0.387599381 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.510426469
## 496 -0.399292629 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 497 -0.386433311 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.273288584
## 498 -0.388900310 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.698295497
## 499 -0.392302024 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.378064199
## 500 -0.399427488 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.018526795
## 501 -0.394015670 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.366678197
## 502 -0.412820431 -0.48724019 0.11562398 -0.2723291 0.15796779 0.438881424
## 503 -0.414838673 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.234315927
## 504 -0.413037834 -0.48724019 0.11562398 -0.2723291 0.15796779 0.983986256
## 505 -0.407360947 -0.48724019 0.11562398 -0.2723291 0.15796779 0.724954717
## 506 -0.414589880 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.362408446
## age dis rad tax ptratio black
## 1 -0.119894767 0.1400749840 -0.9818712 -0.66594918 -1.45755797 0.440615895
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450 0.440615895
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450 0.396035074
## 4 -0.809087830 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.415751408
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.440615895
## 6 -0.350809969 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.410165113
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851 0.426376321
## 8 0.977840575 1.0236248974 -0.5224844 -0.57694801 -1.50374851 0.440615895
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851 0.328123258
## 10 0.615481336 1.3283202075 -0.5224844 -0.57694801 -1.50374851 0.328999540
## 11 0.913894827 1.2117799501 -0.5224844 -0.57694801 -1.50374851 0.392639483
## 12 0.508905089 1.1547920492 -0.5224844 -0.57694801 -1.50374851 0.440615895
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851 0.370513376
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274 0.440615895
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274 0.255720500
## 16 -0.428965883 0.3341187865 -0.6373311 -0.60068166 1.17530274 0.426595391
## 17 -1.395257187 0.3341187865 -0.6373311 -0.60068166 1.17530274 0.330533033
## 18 0.466274590 0.2198105553 -0.6373311 -0.60068166 1.17530274 0.329437681
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274 -0.741378303
## 20 0.032864520 0.0006920764 -0.6373311 -0.60068166 1.17530274 0.375442459
## 21 1.048891406 0.0013569352 -0.6373311 -0.60068166 1.17530274 0.217930861
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274 0.392749018
## 23 0.821528746 0.0863638874 -0.6373311 -0.60068166 1.17530274 0.440615895
## 24 1.116389695 0.1425444597 -0.6373311 -0.60068166 1.17530274 0.414765591
## 25 0.906789743 0.2871037683 -0.6373311 -0.60068166 1.17530274 0.412465352
## 26 0.608376252 0.3132232229 -0.6373311 -0.60068166 1.17530274 -0.583319029
## 27 0.771793164 0.4212152950 -0.6373311 -0.60068166 1.17530274 0.221326452
## 28 0.718505041 0.3126533438 -0.6373311 -0.60068166 1.17530274 -0.550896614
## 29 0.917447368 0.3132707128 -0.6373311 -0.60068166 1.17530274 0.342472368
## 30 0.665216917 0.2108349609 -0.6373311 -0.60068166 1.17530274 0.258020739
## 31 0.906789743 0.2079855659 -0.6373311 -0.60068166 1.17530274 0.038293155
## 32 1.116389695 0.1804414138 -0.6373311 -0.60068166 1.17530274 0.219683424
## 33 0.476932215 0.0925850666 -0.6373311 -0.60068166 1.17530274 -1.359047220
## 34 0.938762618 -0.0037244859 -0.6373311 -0.60068166 1.17530274 0.022958229
## 35 1.006260907 -0.0167367233 -0.6373311 -0.60068166 1.17530274 -1.186967442
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## 427 -0.315284553 -0.8536040479 1.6596029 1.52941294 0.80577843 -3.636831424
## 428 0.359698343 -0.9175729666 1.6596029 1.52941294 0.80577843 -3.700690438
## 429 0.338383094 -0.8830477967 1.6596029 1.52941294 0.80577843 -2.847301799
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843 -3.241738006
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843 -2.992764527
## 432 0.913894827 -0.8105781827 1.6596029 1.52941294 0.80577843 -3.015985986
## 433 0.221149222 -0.7572944954 1.6596029 1.52941294 0.80577843 -2.833938506
## 434 0.686532167 -0.7024911307 1.6596029 1.52941294 0.80577843 -2.809402625
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843 -2.804583076
## 436 0.924552451 -0.7932443629 1.6596029 1.52941294 0.80577843 -2.703591634
## 437 0.878369411 -0.8512295520 1.6596029 1.52941294 0.80577843 -3.605723431
## 438 1.116389695 -0.8932106390 1.6596029 1.52941294 0.80577843 -3.804748865
## 439 0.686532167 -0.9376612017 1.6596029 1.52941294 0.80577843 -3.151590547
## 440 0.899684660 -0.9392758589 1.6596029 1.52941294 0.80577843 0.440615895
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843 0.380919218
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843 0.320784401
## 443 1.116389695 -0.8501847738 1.6596029 1.52941294 0.80577843 0.427362137
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843 0.329218610
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843 -1.272295352
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843 -3.435177145
## 447 0.988498199 -0.8182715493 1.6596029 1.52941294 0.80577843 -0.423507192
## 448 0.995603282 -0.7584342534 1.6596029 1.52941294 0.80577843 0.348825409
## 449 1.070206655 -0.7282306659 1.6596029 1.52941294 0.80577843 0.440615895
## 450 1.055996489 -0.7646079427 1.6596029 1.52941294 0.80577843 -0.574665749
## 451 0.853501620 -0.6987869171 1.6596029 1.52941294 0.80577843 -3.903330533
## 452 1.052443947 -0.6837801032 1.6596029 1.52941294 0.80577843 -0.015160016
## 453 0.825081288 -0.6776064140 1.6596029 1.52941294 0.80577843 0.311254840
## 454 1.091521905 -0.6374774338 1.6596029 1.52941294 0.80577843 0.210263398
## 455 0.906789743 -0.6168668096 1.6596029 1.52941294 0.80577843 -3.833666154
## 456 0.636796585 -0.6455032298 1.6596029 1.52941294 0.80577843 -3.349082489
## 457 0.686532167 -0.5767378294 1.6596029 1.52941294 0.80577843 -3.792042783
## 458 0.416539008 -0.4824228534 1.6596029 1.52941294 0.80577843 -3.868498343
## 459 0.537325421 -0.4805707466 1.6596029 1.52941294 0.80577843 -0.925178346
## 460 0.562193212 -0.5117241325 1.6596029 1.52941294 0.80577843 0.440615895
## 461 0.761135540 -0.5687120333 1.6596029 1.52941294 0.80577843 -1.111169093
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843 0.380700148
## 463 0.512457630 -0.5036983364 1.6596029 1.52941294 0.80577843 0.440615895
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843 0.406879058
## 465 -0.112789684 -0.3949464255 1.6596029 1.52941294 0.80577843 0.440615895
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843 -0.243979021
## 467 0.572850837 -0.4385896596 1.6596029 1.52941294 0.80577843 -3.665748713
## 468 0.920999910 -0.5958762661 1.6596029 1.52941294 0.80577843 -0.278044464
## 469 0.086152643 -0.4210658801 1.6596029 1.52941294 0.80577843 0.132164810
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843 0.440615895
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843 0.440615895
## 472 0.786003330 -0.3304076278 1.6596029 1.52941294 0.80577843 0.423418871
## 473 0.228254306 -0.4267171803 1.6596029 1.52941294 0.80577843 0.401949974
## 474 -0.034633770 -0.5993905200 1.6596029 1.52941294 0.80577843 0.197228711
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843 -0.044844052
## 476 1.024023615 -0.7546350600 1.6596029 1.52941294 0.80577843 -0.590548351
## 477 0.889027036 -0.7074775720 1.6596029 1.52941294 0.80577843 0.433057967
## 478 1.020471073 -0.8046419430 1.6596029 1.52941294 0.80577843 -0.078799960
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843 0.252215374
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843 0.291867112
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843 0.440615895
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843 0.398663919
## 483 0.299305137 -0.1825715149 1.6596029 1.52941294 0.80577843 0.422871195
## 484 -1.004477616 0.1440166471 1.6596029 1.52941294 0.80577843 0.397020891
## 485 -0.947636951 -0.0337381137 1.6596029 1.52941294 0.80577843 0.153962312
## 486 -0.592382795 0.0933923952 1.6596029 1.52941294 0.80577843 0.349920761
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843 0.394392046
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843 0.345539353
## 489 0.857054162 -0.9375187319 -0.6373311 1.79641644 0.75958789 0.420790026
## 490 1.055996489 -0.9686246278 -0.6373311 1.79641644 0.75958789 -0.138277566
## 491 1.045338864 -0.9367114033 -0.6373311 1.79641644 0.75958789 -0.418906714
## 492 1.073759197 -0.9151034909 -0.6373311 1.79641644 0.75958789 0.366241503
## 493 0.530220338 -0.8002728706 -0.6373311 1.79641644 0.75958789 0.440615895
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304 0.440615895
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512 0.34387304 0.440615895
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512 0.34387304 0.401073693
## 497 0.153650933 -0.4732098094 -0.4076377 -0.10227512 0.34387304 0.440615895
## 498 0.071942477 -0.4285217972 -0.4076377 -0.10227512 0.34387304 0.440615895
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512 0.34387304 0.440615895
## 500 0.174966182 -0.6625521101 -0.4076377 -0.10227512 0.34387304 0.428238419
## 501 0.395223759 -0.6158695213 -0.4076377 -0.10227512 0.34387304 0.440615895
## 502 0.018654354 -0.6251775451 -0.9818712 -0.80241764 1.17530274 0.386834118
## 503 0.288647512 -0.7159307773 -0.9818712 -0.80241764 1.17530274 0.440615895
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274 0.440615895
## 505 0.736267749 -0.6677760011 -0.9818712 -0.80241764 1.17530274 0.402826256
## 506 0.434301716 -0.6126402069 -0.9818712 -0.80241764 1.17530274 0.440615895
## lstat medv
## 1 -1.074498970 0.159527789
## 2 -0.491952525 -0.101423917
## 3 -1.207532413 1.322937477
## 4 -1.360170785 1.181588636
## 5 -1.025486649 1.486032293
## 6 -1.042290874 0.670558212
## 7 -0.031236706 0.039924923
## 8 0.909799859 0.496590409
## 9 2.419379350 -0.655946292
## 10 0.622727693 -0.394994586
## 11 1.091845624 -0.819041108
## 12 0.086392864 -0.394994586
## 13 0.428078760 -0.090550930
## 14 -0.615183504 -0.231899770
## 15 -0.335113097 -0.471105500
## 16 -0.585776111 -0.286264709
## 17 -0.850442645 0.061670899
## 18 0.282442149 -0.547216415
## 19 -0.134862757 -0.253645746
## 20 -0.192277190 -0.471105500
## 21 1.171665689 -0.971262937
## 22 0.164812578 -0.318883672
## 23 0.849584722 -0.797295133
## 24 1.012025558 -0.873406047
## 25 0.510699530 -0.753803182
## 26 0.540106923 -0.938643973
## 27 0.302047077 -0.645073304
## 28 0.647934029 -0.840787084
## 29 0.020576319 -0.449359525
## 30 -0.094252548 -0.166661844
## 31 1.392921311 -1.069119826
## 32 0.054184768 -0.873406047
## 33 2.108501199 -1.014754888
## 34 0.797771697 -1.025627875
## 35 1.076441751 -0.982135924
## 36 -0.416333515 -0.394994586
## 37 -0.174072614 -0.275391721
## 38 -0.543765550 -0.166661844
## 39 -0.353317674 0.235638703
## 40 -1.166922204 0.898890955
## 41 -1.494604580 1.344683452
## 42 -1.094103899 0.442225470
## 43 -0.958269752 0.300876629
## 44 -0.730012370 0.235638703
## 45 -0.434538092 -0.144915868
## 46 -0.342114857 -0.351502635
## 47 0.209623843 -0.275391721
## 48 0.860787538 -0.645073304
## 49 2.542610329 -0.884279035
## 50 0.496696010 -0.340629648
## 51 0.111599201 -0.308010684
## 52 -0.451342316 -0.221026782
## 53 -1.032488409 0.268257666
## 54 -0.591377519 0.094289862
## 55 0.300646725 -0.394994586
## 56 -1.098304955 1.399048391
## 57 -0.963871160 0.235638703
## 58 -1.218735230 0.985874857
## 59 -0.811232788 0.083416874
## 60 -0.480749709 -0.318883672
## 61 0.069588640 -0.416740562
## 62 0.250234052 -0.710311231
## 63 -0.829437365 -0.036185991
## 64 -0.441539852 0.268257666
## 65 -0.644590896 1.138096685
## 66 -1.117909883 0.105162850
## 67 -0.337913801 -0.340629648
## 68 -0.637589136 -0.057931966
## 69 0.061186528 -0.558089402
## 70 -0.540964846 -0.177534831
## 71 -0.830837717 0.181273764
## 72 -0.388326475 -0.090550930
## 73 -0.998879961 0.029051936
## 74 -0.716008850 0.094289862
## 75 -0.822435605 0.170400776
## 76 -0.519959566 -0.123169893
## 77 -0.095652900 -0.275391721
## 78 -0.333712745 -0.188407819
## 79 -0.043839875 -0.144915868
## 80 -0.497553933 -0.242772758
## 81 -1.031088057 0.594447298
## 82 -0.760820115 0.148654801
## 83 -0.830837717 0.246511690
## 84 -0.720209906 0.039924923
## 85 -0.424735627 0.148654801
## 86 -0.857444405 0.442225470
## 87 0.028978431 -0.003567028
## 88 -0.589977167 -0.036185991
## 89 -1.001680665 0.116035838
## 90 -0.973673624 0.670558212
## 91 -0.538164142 0.007305960
## 92 -0.623585616 -0.057931966
## 93 -0.629187024 0.039924923
## 94 -0.902255670 0.268257666
## 95 -0.288901480 -0.210153795
## 96 -0.840640181 0.637939249
## 97 -0.183875078 -0.123169893
## 98 -1.182326077 1.757856987
## 99 -1.271948607 2.312379361
## 100 -0.905056374 1.159842661
## 101 -0.452742668 0.540082359
## 102 -0.697804274 0.431352482
## 103 -0.283300072 -0.427613550
## 104 0.110198849 -0.351502635
## 105 -0.045240227 -0.264518733
## 106 0.534505515 -0.329756660
## 107 0.841182610 -0.329756660
## 108 0.201221731 -0.231899770
## 109 -0.053642339 -0.297137697
## 110 0.405673128 -0.340629648
## 111 0.048583360 -0.090550930
## 112 -0.349116618 0.029051936
## 113 0.498096362 -0.405867574
## 114 0.621327341 -0.416740562
## 115 -0.308506409 -0.438486537
## 116 0.435080520 -0.460232513
## 117 -0.085850436 -0.144915868
## 118 -0.329511689 -0.362375623
## 119 0.380466791 -0.231899770
## 120 0.134004834 -0.351502635
## 121 0.240431588 -0.057931966
## 122 0.226428068 -0.242772758
## 123 0.738956911 -0.221026782
## 124 1.786420232 -0.568962390
## 125 0.689944590 -0.405867574
## 126 0.302047077 -0.123169893
## 127 2.045485358 -0.742930194
## 128 0.635330861 -0.688565255
## 129 0.383267495 -0.492851476
## 130 0.796371345 -0.895152022
## 131 -0.007430722 -0.362375623
## 132 -0.055042691 -0.318883672
## 133 -0.214682823 0.050797911
## 134 0.332854822 -0.449359525
## 135 0.652135085 -0.753803182
## 136 0.603122764 -0.481978488
## 137 0.594720652 -0.558089402
## 138 0.271239333 -0.590708366
## 139 1.213676250 -1.003881900
## 140 0.813175569 -0.514597451
## 141 1.611376228 -0.927770986
## 142 3.046737061 -0.884279035
## 143 1.983869868 -0.993008912
## 144 1.927855787 -0.753803182
## 145 2.329756820 -1.166976716
## 146 2.121104367 -0.949516961
## 147 0.559711851 -0.753803182
## 148 2.363365269 -0.862533059
## 149 2.193922673 -0.514597451
## 150 1.231880827 -0.775549157
## 151 0.202622083 -0.112296905
## 152 0.087793216 -0.318883672
## 153 -0.074647619 -0.786422145
## 154 0.439281577 -0.340629648
## 155 0.345457990 -0.601581353
## 156 0.331454470 -0.753803182
## 157 0.488293898 -1.025627875
## 158 -1.129112700 2.040554668
## 159 -0.871447926 0.192146752
## 160 -0.737014131 0.083416874
## 161 -1.001680665 0.485717421
## 162 -1.529613381 2.986504601
## 163 -1.503006692 2.986504601
## 164 -1.306957408 2.986504601
## 165 -0.141864517 0.018178948
## 166 -0.398128939 0.268257666
## 167 -1.253744030 2.986504601
## 168 -0.071846915 0.137781813
## 169 -0.217483527 0.137781813
## 170 -0.186675782 -0.025313003
## 171 0.248833700 -0.558089402
## 172 -0.087250788 -0.373248611
## 173 0.285242853 0.061670899
## 174 -0.505956045 0.116035838
## 175 -0.421934923 0.007305960
## 176 -1.025486649 0.746669127
## 177 -0.356118378 0.072543887
## 178 -0.891052854 0.224765715
## 179 -0.802830676 0.801034065
## 180 -1.066096858 1.594762170
## 181 -0.713208146 1.877459852
## 182 -0.448541612 1.486032293
## 183 -1.096904603 1.670873085
## 184 -0.976474328 1.083731747
## 185 0.185817859 0.420479494
## 186 0.069588640 0.768415102
## 187 -1.148717628 2.986504601
## 188 -0.836439125 1.029366808
## 189 -1.133313756 0.790161078
## 190 -1.017084537 1.344683452
## 191 -1.057694746 1.573016195
## 192 -1.115109179 0.866271992
## 193 -1.369973249 1.507778268
## 194 -1.067497210 0.931509918
## 195 -1.158520092 0.714050163
## 196 -1.355969729 2.986504601
## 197 -1.200530653 1.170715648
## 198 -0.566171183 0.844526016
## 199 -0.844841237 1.312064489
## 200 -1.133313756 1.344683452
## 201 -1.148717628 1.127223698
## 202 -0.731412722 0.170400776
## 203 -1.336364800 2.149284545
## 204 -1.238340158 2.823409785
## 205 -1.368572897 2.986504601
## 206 -0.249691623 0.007305960
## 207 -0.235688103 0.203019739
## 208 0.757161488 -0.003567028
## 209 0.281041797 0.203019739
## 210 1.461538560 -0.275391721
## 211 0.646533677 -0.090550930
## 212 1.586169891 -0.351502635
## 213 0.472890025 -0.014440015
## 214 -0.458344076 0.605320286
## 215 2.366165973 0.126908825
## 216 -0.445740908 0.268257666
## 217 0.120001313 0.083416874
## 218 -0.414933163 0.670558212
## 219 0.737556559 -0.112296905
## 220 -0.301504649 0.050797911
## 221 -0.412132459 0.453098458
## 222 1.233281179 -0.090550930
## 223 -0.381324714 0.540082359
## 224 -0.707606738 0.822780041
## 225 -1.192128541 2.421109239
## 226 -1.123511291 2.986504601
## 227 -1.333564096 1.638254121
## 228 -0.881250390 0.985874857
## 229 -1.222936286 2.627696006
## 230 -1.245341918 0.975001869
## 231 -0.140464165 0.192146752
## 232 -1.036689465 0.996747845
## 233 -1.425987330 2.084046619
## 234 -1.218735230 2.801663810
## 235 -0.644590896 0.703177176
## 236 -0.248291271 0.159527789
## 237 -0.435938444 0.279130654
## 238 -1.109507771 0.975001869
## 239 -0.881250390 0.126908825
## 240 -0.739814835 0.083416874
## 241 -0.178273670 -0.057931966
## 242 -0.035437762 -0.264518733
## 243 -0.200679302 -0.036185991
## 244 -1.045091578 0.126908825
## 245 -0.021434242 -0.536343427
## 246 0.813175569 -0.438486537
## 247 -0.489151821 0.192146752
## 248 -0.350516970 -0.221026782
## 249 -0.438739148 0.213892727
## 250 -0.853243349 0.398733519
## 251 -0.945666583 0.203019739
## 252 -1.269147903 0.246511690
## 253 -1.277550015 0.768415102
## 254 -1.276149663 2.203649484
## 255 -0.851842997 -0.068804954
## 256 -0.476548653 -0.177534831
## 257 -1.336364800 2.334125337
## 258 -1.054894042 2.986504601
## 259 -0.681000049 1.464286318
## 260 -0.805631380 0.822780041
## 261 -0.428936683 1.225080587
## 262 -0.755218707 2.236268447
## 263 -0.944266231 2.856028748
## 264 -0.196478246 0.920636930
## 265 -0.637589136 1.518651256
## 266 -0.308506409 0.029051936
## 267 0.299246373 0.888017967
## 268 -0.730012370 2.986504601
## 269 -1.329363040 2.279760398
## 270 0.139606242 -0.199280807
## 271 0.048583360 -0.155788856
## 272 -0.849042293 0.290003641
## 273 -0.689402161 0.203019739
## 274 -0.850442645 1.377302416
## 275 -1.277550015 1.072858759
## 276 -1.354569377 1.029366808
## 277 -0.924661303 1.159842661
## 278 -1.189327837 1.148969673
## 279 -0.765021171 0.714050163
## 280 -1.092703547 1.366429428
## 281 -1.245341918 2.486347165
## 282 -1.129112700 1.399048391
## 283 -1.350368321 2.551585092
## 284 -1.329363040 2.986504601
## 285 -0.672597937 1.051112783
## 286 -0.619384560 -0.057931966
## 287 0.038780895 -0.264518733
## 288 -0.772022931 0.072543887
## 289 -0.707606738 -0.025313003
## 290 -0.440139500 0.246511690
## 291 -1.305557056 0.648812237
## 292 -1.273348959 1.605635158
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## 400 2.424980758 -1.764991042
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## 404 0.996621685 -1.547531287
## 405 2.062289582 -1.525785311
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## 419 1.115651608 -1.493166348
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## 429 1.241683291 -1.253960618
## 430 1.600173411 -1.417055434
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## 432 0.985418869 -0.916897998
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## 437 0.755761136 -1.406182446
## 438 1.932056843 -1.504039336
## 439 2.992123331 -1.536658299
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## 442 0.961612885 -0.590708366
## 443 0.551309739 -0.449359525
## 444 0.867789298 -0.775549157
## 445 1.559563202 -1.275706593
## 446 1.586169891 -1.166976716
## 447 0.719351983 -0.829914096
## 448 0.530304459 -1.079992814
## 449 0.766963952 -0.916897998
## 450 0.932205492 -1.036500863
## 451 0.670339662 -0.993008912
## 452 0.710949871 -0.797295133
## 453 0.646533677 -0.699438243
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## 455 0.848184370 -0.829914096
## 456 0.766963952 -0.916897998
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## 466 0.206823139 -0.286264709
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## 468 1.213676250 -0.373248611
## 469 0.766963952 -0.373248611
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## 471 0.509299178 -0.286264709
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## 477 0.843983314 -0.634200317
## 478 1.716402630 -1.145230740
## 479 0.752960432 -0.862533059
## 480 0.063987232 -0.123169893
## 481 -0.267896200 0.050797911
## 482 -0.688001809 0.126908825
## 483 -0.790227508 0.268257666
## 484 -0.312707465 -0.079677942
## 485 0.096195329 -0.210153795
## 486 -0.290301832 -0.144915868
## 487 0.325853062 -0.373248611
## 488 -0.168471205 -0.210153795
## 489 0.757161488 -0.797295133
## 490 1.584769539 -1.688880128
## 491 2.384370549 -1.569277262
## 492 0.758561840 -0.971262937
## 493 0.097595681 -0.264518733
## 494 -0.090051492 -0.079677942
## 495 0.131204129 0.213892727
## 496 0.692745294 0.061670899
## 497 1.188469914 -0.308010684
## 498 0.202622083 -0.460232513
## 499 0.037380543 -0.144915868
## 500 0.342657286 -0.547216415
## 501 0.234830180 -0.623327329
## 502 -0.417733867 -0.014440015
## 503 -0.500354637 -0.210153795
## 504 -0.982075736 0.148654801
## 505 -0.864446166 -0.057931966
## 506 -0.668396881 -1.156103728boston_scaled <- as.data.frame(scale(Boston))
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610#Create a categorical variable of the crime rate
crime <- cut(boston_scaled$crim,
breaks = bins,
include.lowest = TRUE,
label = c("low", "medium_low", "medium_high", "high"))
summary(crime)## low medium_low medium_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)Here begins the splitting of the data into a training set and a test set, with a split ratio of 80% of the training set data and 20% of the test set data.
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- boston_scaled$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)
summary(train)## zn indus chas nox
## Min. :-0.48724 Min. :-1.556302 Min. :-0.27233 Min. :-1.464433
## 1st Qu.:-0.48724 1st Qu.:-0.866833 1st Qu.:-0.27233 1st Qu.:-0.912126
## Median :-0.48724 Median :-0.293247 Median :-0.27233 Median :-0.144075
## Mean :-0.01761 Mean :-0.009373 Mean :-0.01895 Mean : 0.005272
## 3rd Qu.: 0.04872 3rd Qu.: 1.014995 3rd Qu.:-0.27233 3rd Qu.: 0.598087
## Max. : 3.80047 Max. : 2.420170 Max. : 3.66477 Max. : 2.729645
## rm age dis rad
## Min. :-3.055198 Min. :-2.33313 Min. :-1.265817 Min. :-0.98187
## 1st Qu.:-0.566645 1st Qu.:-0.86504 1st Qu.:-0.807242 1st Qu.:-0.63733
## Median :-0.128995 Median : 0.29931 Median :-0.246540 Median :-0.52248
## Mean : 0.002574 Mean :-0.00529 Mean : 0.009968 Mean :-0.02671
## 3rd Qu.: 0.422870 3rd Qu.: 0.90679 3rd Qu.: 0.683597 3rd Qu.: 0.28144
## Max. : 3.551530 Max. : 1.11639 Max. : 3.956602 Max. : 1.65960
## tax ptratio black lstat
## Min. :-1.31269 Min. :-2.7047 Min. :-3.87923 Min. :-1.529613
## 1st Qu.:-0.77868 1st Qu.:-0.6723 1st Qu.: 0.20591 1st Qu.:-0.803531
## Median :-0.46718 Median : 0.2746 Median : 0.38256 Median :-0.194378
## Mean :-0.02849 Mean :-0.0109 Mean : 0.01762 Mean :-0.001739
## 3rd Qu.: 1.52941 3rd Qu.: 0.8058 3rd Qu.: 0.43350 3rd Qu.: 0.577916
## Max. : 1.79642 Max. : 1.6372 Max. : 0.44062 Max. : 3.545262
## medv crime
## Min. :-1.9063399 low :103
## 1st Qu.:-0.5825536 medium_low : 96
## Median :-0.1231699 medium_high:108
## Mean :-0.0006066 high : 97
## 3rd Qu.: 0.2682577
## Max. : 2.9865046summary(test)## zn indus chas nox
## Min. :-0.48724 Min. :-1.51549 Min. :-0.27233 Min. :-1.42991
## 1st Qu.:-0.48724 1st Qu.:-0.89526 1st Qu.:-0.27233 1st Qu.:-0.90997
## Median :-0.48724 Median :-0.18028 Median :-0.27233 Median :-0.16996
## Mean : 0.06974 Mean : 0.03713 Mean : 0.07506 Mean :-0.02088
## 3rd Qu.: 0.28991 3rd Qu.: 1.01499 3rd Qu.:-0.27233 3rd Qu.: 0.65850
## Max. : 3.58609 Max. : 2.42017 Max. : 3.66477 Max. : 2.72965
## rm age dis rad
## Min. :-3.87641 Min. :-2.21589 Min. :-1.24278 Min. :-0.9819
## 1st Qu.:-0.57376 1st Qu.:-0.79488 1st Qu.:-0.80354 1st Qu.:-0.6373
## Median :-0.09128 Median : 0.36325 Median :-0.40015 Median :-0.5225
## Mean :-0.01020 Mean : 0.02095 Mean :-0.03948 Mean : 0.1058
## 3rd Qu.: 0.65308 3rd Qu.: 0.88636 3rd Qu.: 0.62572 3rd Qu.: 1.6596
## Max. : 2.81002 Max. : 1.11639 Max. : 2.57645 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3068 Min. :-2.51994 Min. :-3.90333 Min. :-1.369973
## 1st Qu.:-0.6659 1st Qu.:-0.48756 1st Qu.: 0.16459 1st Qu.:-0.786377
## Median :-0.1438 Median : 0.27459 Median : 0.37703 Median :-0.049441
## Mean : 0.1128 Mean : 0.04318 Mean :-0.06981 Mean : 0.006889
## 3rd Qu.: 1.5294 3rd Qu.: 0.80578 3rd Qu.: 0.42695 3rd Qu.: 0.628679
## Max. : 1.7964 Max. : 1.26768 Max. : 0.44062 Max. : 3.406628
## medv
## Min. :-1.688880
## 1st Qu.:-0.699438
## Median :-0.242773
## Mean : 0.002403
## 3rd Qu.: 0.540082
## Max. : 2.986505nrow(train)## [1] 404nrow(test)## [1] 1024.Fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables.
# linear discriminant analysis
lda.fit <- lda(crime ~., data = train)
# print the lda.fit object
lda.fit ## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low medium_low medium_high high
## 0.2549505 0.2376238 0.2673267 0.2400990
##
## Group means:
## zn indus chas nox rm age
## low 0.9031446 -0.8892919 -0.1194320 -0.8552617 0.39404213 -0.8827600
## medium_low -0.1107253 -0.3477268 -0.0262603 -0.6056877 -0.14656366 -0.3763069
## medium_high -0.3911636 0.2085357 0.1651265 0.4152634 0.07405495 0.4320649
## high -0.4872402 1.0172187 -0.1099744 1.0672096 -0.34509468 0.8066960
## dis rad tax ptratio black
## low 0.8630832 -0.6941969 -0.7406065 -0.42208269 0.37714291
## medium_low 0.4180879 -0.5476071 -0.5040165 -0.03094393 0.30764931
## medium_high -0.3942125 -0.4214618 -0.3114279 -0.31092061 0.08275328
## high -0.8498137 1.6371072 1.5133254 0.77958792 -0.72368103
## lstat medv
## low -0.75472926 0.47801132
## medium_low -0.14292937 -0.02066933
## medium_high 0.02668341 0.16617239
## high 0.90591641 -0.67466577
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.11088916 0.647153102 -1.13656278
## indus -0.02360436 -0.246326830 0.16167930
## chas -0.10161839 -0.049986355 0.08562074
## nox 0.39635377 -0.736510328 -1.16948265
## rm -0.11489520 -0.045214748 -0.11927465
## age 0.22561598 -0.355937823 -0.07269040
## dis -0.12037413 -0.244199472 0.40951042
## rad 3.12585185 0.919048722 -0.13504872
## tax 0.04429596 0.014310552 0.61423702
## ptratio 0.13958790 0.002205165 -0.25280194
## black -0.12121180 0.022033038 0.07561078
## lstat 0.23186553 -0.170514627 0.38361382
## medv 0.20593422 -0.382529352 -0.15318653
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9464 0.0396 0.0140- Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data.
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
In this graph, the clustering of crime levels can be clearly seen, with
low and medium_low crime rates concentrated in the top left and middle
left; medium_high and high crime rates are concentrated in the bottom
left and right. The concentration of low crime is much greater, and
there is a very clear demarcation line between high crime on the right
and the left. In the red part of the graph, rad(index of accessibility
to radial highway) is associated with high crime rates and zn(proportion
of residential land zoned for lots over 25,000 sq.ft) is closely related
to low crime rates.
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
predictions <- lda.pred$class
#table(correct = correct_classes, predicted = predictions)6.Calculate the distances between the observations. Run k-means algorithm on the dataset.
library(MASS)
#reload the Boston dataset and standardize the detaset
data("Boston")
boston_scaled <- as.data.frame(scale(Boston))
#calculate the distances between the observations
dist <- dist(boston_scaled)
summary(dist)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970k-means
set.seed(13)
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
km <- kmeans(Boston, centers = 4)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
pairs(Boston[1:5], col = km$cluster)
pairs(Boston[11:14], col = km$cluster)
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <- kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
The optimal number of clusters can be found by looking at the sum of
squares within clusters, and in the graph above I think the optimal
number of clusters is close to 2. At an optimal cluster number of 2, the
values vary significantly.
Prepare some package
library(dplyr)
library(tidyverse)
library(GGally)Read data
human <- read.table("human.txt", sep = ",", )dim(human)## [1] 155 8The dataset has 155 observations and 8 variables.
- Show a graphical overview of the data and show summaries of the variables in the data.
ggpairs(human)
summary(human)## edu_fm labour_fm edu.exp life.exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## gni maternal.mortality.ratio teen.birth per.pa
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50The correlation between the variables can be seen very clearly in the visual charts. Take the variable of the labour market participation of females and males (labour_fm) as an example, there is a strong correlation between it and the percentage of parliamentary representation (per.pa), maternal mortality rate (maternal.mortality.ratio). As can be seen from the scatter plot, there appears to be a positive correlation between labour_fm and per.pa and maternal.mortality.ratio.
# Access corrplot
library(corrplot)
# compute the correlation matrix and visualize it with corrplot
cor(human) %>%
corrplot(method = "number")
In a heat map, the degree of correlation and correlations between
variables can be seen more visually through the figures.
2-3Perform principal component analysis (PCA) on the raw (non-standardized) human data.
# standardize the variables
human_std <- scale(human)
# print out summaries of the standardized variables
summary(human_std)## edu_fm labour_fm edu.exp life.exp
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## gni maternal.mortality.ratio teen.birth per.pa
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850# perform principal component analysis (with the SVD method)
pca_human <- prcomp(human)
pca_human_std <- prcomp(human_std)
# print out summaries of the standardized variables
summary(pca_human_std)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.071 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.536 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion 0.536 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
## PC8
## Standard deviation 0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion 1.00000summary(pca_human)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912 0.1591
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000 1.0000# draw a biplot of the principal component representation and the original variables
biplot(pca_human,
choices = 1:2,
cex = c(0.8, 1),
col = c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
biplot(pca_human_std,
choices = 1:2,
cex = c(0.8, 1),
col = c("grey40", "deeppink2"))
4.Give personal interpretations of the first two principal component
dimensions based on the biplot drawn after PCA on the standardized human
data
In this figure, the countries on the left are more concentrated, with an increase in educational expectations, life expectancy and the level of primary education attained by both men and women in this region, with a number of European or East Asian countries. In contrast, the number of women in parliament and the ratio of men to women in the labour market are concentrated at the top of the graph, mostly in the Nordic countries. Finally, the high maternal mortality and adolescent birth rates point to the countries on the right, which are mostly African countries.
- Tea data
#read data
tea <- read.csv("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/tea.csv", stringsAsFactors = TRUE)#explore the data
str(tea); dim(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "+60","15-24",..: 4 5 5 2 5 2 4 4 4 4 ...
## $ frequency : Factor w/ 4 levels "+2/day","1 to 2/week",..: 3 3 1 3 1 3 4 2 1 1 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...## [1] 300 36Tea dataset has 300 observations and 36 varibales, and all the variables are categorical.
#convert all variables to factors
tea$Tea <- factor(tea$Tea)
tea$How <- factor(tea$How)
tea$how <- factor(tea$how)
tea$sugar <- factor(tea$sugar)
tea$where <- factor(tea$where)
tea$lunch <- factor(tea$lunch)library(dplyr)
library(tidyr)
# column names to keep in the dataset
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
# select the 'keep_columns' to create a new dataset
tea_time <- dplyr::select(tea, keep_columns)## Warning: Using an external vector in selections was deprecated in tidyselect 1.1.0.
## ℹ Please use `all_of()` or `any_of()` instead.
## # Was:
## data %>% select(keep_columns)
##
## # Now:
## data %>% select(all_of(keep_columns))
##
## See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.# look at the summaries and structure of the data
summary(tea_time)## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
## str(tea_time)## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...# visualize the dataset
library(ggplot2)
pivot_longer(tea_time, cols = everything()) %>%
ggplot(aes(value)) +
facet_wrap("name", scales = "free") +
geom_bar() +
theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
In tea dataset, I choose 6 variables, The frequency distribution chart
gives a very visual indication of the number of times each of the six
variables was answered.
# multiple correspondence analysis
library(FactoMineR)
mca <- MCA(tea_time, graph = FALSE)
# summary of the model
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.279 0.261 0.219 0.189 0.177 0.156 0.144
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519 7.841
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953 77.794
## Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.141 0.117 0.087 0.062
## % of var. 7.705 6.392 4.724 3.385
## Cumulative % of var. 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139 0.003
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626 0.027
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111 0.107
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841 0.127
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979 0.035
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990 0.020
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347 0.102
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459 0.161
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968 0.478
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898 0.141
## v.test Dim.3 ctr cos2 v.test
## black 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 2.867 | 0.433 9.160 0.338 10.053 |
## green -5.669 | -0.108 0.098 0.001 -0.659 |
## alone -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 3.226 | 1.329 14.771 0.218 8.081 |
## milk 2.422 | 0.013 0.003 0.000 0.116 |
## other 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |# visualize MCA
plot(mca, invisible=c("ind"), graph.type = "classic", habillage = "quali")
In the Mac factor map, different colours indicate different variables.
Red indicates how the tea is drunk, with lemon, milk, other or nothing.
Green indicates how the tea is packaged. Pink indicates where the tea
was purchased. Yellow indicates whether the tea was consumed before or
after lunch. Black indicates the variety of tea. Blue indicates whether
or not sugar was added. In the map we can observe that Earl Grey tea is
usually served with milk and sugar and is usually served at breakfast
time; black tea is served with lemon; green tea is usually bought
unpackaged from a tea shop and is usually served at breakfast or after
lunch.